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Question

Suppose that a and b are two positive real numbers such that log27a+log9b=72 and log27b+log9a=23, then the value of product ab is

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Solution

log27a+log9b=72
log33a+log32b=72
13log3a+12log3b=72
log3a1/3+log3b1/2=72
log3(a1/3b1/2)=72
a1/3b1/2=37/2 ...(1)

log27b+log9a=23
log33b+log32a=23
13log3b+12log3a=23
log3b1/3+log3a1/2=23
log3(b1/3a1/2)=23
a1/2b1/3=32/3 ...(2)

Multiplying eqn (1) and (2), we get
a5/6b5/6=325/6
(ab)5/6=325/6
ab=35=243

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